A gravitational action with stringy $Q$ and $R$ fluxes via deformed differential graded Poisson algebras

Abstract

We study a deformation of a $2$-graded Poisson algebra where the functions of the phase space variables are complemented by linear functions of parity odd velocities. The deformation is carried by a $2$-form $B$-field and a bivector $\Pi$, that we consider as gauge fields of the geometric and non-geometric fluxes $H$, $f$, $Q$ and $R$ arising in the context of string theory compactification. The technique used to deform the Poisson brackets is widely known for the point particle interacting with a $U(1)$ gauge field, but not in the case of non-abelian or higher spin fields. The construction is closely related to Generalized Geometry: With an element of the algebra that squares to zero, the graded symplectic picture is equivalent to an exact Courant algebroid over the generalized tangent bundle $E\cong TM\oplus T^{\ast }M$, and to its higher gauge theory. A particular idempotent graded canonical transformation is equivalent to the generalized metric. Focusing on the generalized differential geometry side we construct an action functional with the Ricci tensor of a connection on covectors, encoding the dynamics of a gravitational theory for a contravariant metric tensor and $Q$ and $R$ fluxes. We also extract a connection on vector fields and determine a non-symmetric metric gravity theory involving a metric and $H$-flux.

27 pages

References in corpus (6)

• Flux Compactification
• Non-geometric Backgrounds and the First Order String Sigma Model
• A bi-invariant Einstein-Hilbert action for the non-geometric string
• Moduli Stabilisation and Applications in IIB String Theory
• Letters to Alan Weinstein about Courant algebroids
• Unified picture of non-geometric fluxes and T-duality in double field theory via graded symplectic manifolds